Step by step
The question
Question: find (f/g)(x) for f(x) = x + 5 and g(x) = 3x.
(f/g)(x) means f of x divided by g of x.
We write it as a fraction: f on top, g at the bottom.
1 Put the rules in
Each rule goes in as one piece: (x + 5)/(3x).
2 Find what x can't be
We can never divide by 0.
So the bottom, g of x, can never be 0.
Careful!
The answer is not only the fraction.
x = 0 makes the bottom 0, so we must say it is not allowed.
Which x makes 3x zero?
We solve 3x = 0.
3 Remove the 3
3x means 3 times x.
We want just x.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3 divided by 3 is 1.
So just x is left.
0 divided by 3 is 0.
So x = 0!
So (f/g)(x) = (x + 5)/(3x), with x ≠ 0.
4 Check
Let's check with a number: x = 1.
f of 1 is 6, g of 1 is 3.
Divided: 2.
That is our fraction at x = 1.
It works!
Let's look back.
First: put the rules in.
Then: find what x can't be.
Then: remove the 3.
Last: check.